Exponents and Roots

Exponents are used when you multiply a number by itself

the exponent can be found raised to the upper right of the base number

all exponents can be expressed as a number to the power of another number

22 = 2 x 2 = 4                  this can also be expressed as: two squared

23 = 2 x 2 x 2 = 8            this can also be expressed as: two cubed 

34 = 3 x 3 x 3 x 3 = 81   this is expressed as: three to the power of four

Please note:  any number to the power of 0 is 1        ( 20 = 1 )

                    any number to the power of 1 is itself        ( 21 = 2 )

Here is a story that illustrates both the idea of exponents and the BEDMAS order of operations

Two men were walking in the woods when they met two more men and now there were four men.  The four men found a mathematical machine.  They stepped into the machine and were squared.  There were now sixteen men.  When that happened some of them got scared so four of the sixteen men decided to leave.  This left twelve men.  The twelve remaining men thought that was fun so they decided to do it again to see what would happen.  When the twelve men stepped back into the machine and came back out they found that they had doubled to twenty-four.  By this time it was getting late in the day so they decided to go back into the machine.  When the twenty-four men stepped back into the machine instead of multiplying they divided by twelve and came out as the original two men walking in the woods once again.

                                                                     [ ( 2 + 2 )2 - 4 ] x 2 ÷ 12

Roots (sometimes known as radicals) can be seen as the opposite of exponents

Square roots are the used most often.  The answer of a square root can be determined by 

finding what two identical numbers multiply with each other to create the number within the square root.

The square root of 4 is 2 because 2 x 2 or 22 = 4

The square root of 9 is 3 because 3 x 3 or 32 = 9

The square root of 16 is 4 because 4 x 4 or 42 = 16

The square root of 25 is 5 because 5 x 5 or 52 = 25 

The square root of 36 is 6 because 6 x6 or 62 = 36

The square root of 49 is 7 because 7 x7 or 72  = 49

The square root of 64 is 8 because 8 x 8 or 82 = 64

           Root three ~ a mathematic love poem 

I fear that I will always be... a lonely number like root three

A three is all that’s good and right… why must my three keep out of sight 

Beneath a vicious square root sign… I wish instead I were a nine 

For nine could thwart this evil trick… with just some quick arithmetic 

I know I’ll never see the sum as 1.7321 

Such is my reality… a sad irrationality 

But hark! What’s this I see... another square root of three

has quietly come waltzing by. Together now, we multiply. 

To form a number we prefer. Rejoicing as an integer… 

We break free of our mortal bonds… with a wave of magic wands. 

Our square root signs become unglued… and Love for me has been renewed.

                                                                          ~ Kumar Patel 

However, we will also encounter cubed, fourth, fifth roots, and so on...   

Examples are illustrated here...

This video is a guide in simplifying an equation that contains roots 

SCIENTIFIC NOTATION

scientific notation uses exponents with a base of 10

(this may also be seen in certain scientific calculators

      ... in order to express large digit numbers)

30000 = 3 x 10000 = 3 x 104

200 = 2 x 100 = 2 x102

0.005 = 5 x 0.001 = 5 x 10-3

100 = 1 x 102

4321 = 4.321 x 103

1.23 = 1.23 x 100

0.25 = 2.5 x 10-1

0.0007925 = 7.925 x 10-4

Further hints...

negative exponents